Physics 11. Unit 7 (Part 1) Wave Motion

Size: px

Start display at page:

Download "Physics 11. Unit 7 (Part 1) Wave Motion"

Morgan Greene
5 years ago
Views:

1 Physics 11 Unit 7 (Part 1) Wave Motion

1. Introduction to wave Wave motion is a popular

For example, light waves, sound waves, radio waves,

It is one the major ways that energy is transmitted

2 1. Introduction to wave Wave motion is a popular phenomenon that we observe often in our daily lives. For example, light waves, sound waves, radio waves, water waves, earthquake waves, and brain waves. It is one the major ways that energy is transmitted from one place to the other. Unit 7 (Part I) - Wave Motion 2

3 How to define a wave? In physics, a wave is an oscillation or vibration that travels through a medium, either mass or space, in which energy is transferred from one place to another. This definition is somehow confusing and inaccurate because not all vibration results in wave motion. Therefore, it may be more appropriate, and more straightforward, to define wave motion as a process in which disturbance (in whatever form it takes) moves away from the source (called propagation) through a medium accompanied by energy transfer. Depending on how it is initiated and how it behaves, wave can be classified into many categories. Unit 7 (Part I) - Wave Motion 3

4 (1) The repetitive nature If a wave is non-repeating, it is called a pulse. (e.g. electrocardiogram, ECG) If a wave recurs at regular time intervals, it is called a periodic wave. Unit 7 (Part I) - Wave Motion 4

5 More precisely, non-repeating wave (aperiodic) can be divided into 2 types: a pulse (a transient wave with a recognizable waveform) or a noise (a random, continuous wave). On the other hand, periodic waves can be categorized into either a simple wave (a single wave) or a complex wave (a mixture of waves). Unit 7 (Part I) - Wave Motion 5

If a wave is produced by the oscillations of the electric and magnetic fields, it is

6 (2) Dependence upon medium If a wave is formed from the oscillation of matter, it is called a mechanical wave. Its transmission does require a medium. If a wave is produced by the oscillations of the electric and magnetic fields, it is called an electromagnetic wave. Its propagation does not require a medium. Unit 7 (Part I) - Wave Motion 6

7 (3) The mode of vibration If the disturbance is at the right angle to the direction that the wave travels, the wave is called a transverse wave. If the disturbance is in the same direction as the direction that the wave travels, the wave is called a longitudinal wave. Unit 7 (Part I) - Wave Motion 7

8 There are several features which are associated with waves and wave motion. (1) Wavelength (λ) Wavelength is the length of one complete wave, which is defined as the distance that must be followed along the wave before it begins repeating itself. For example, for a transverse wave, wavelength is the distance between two successive crests (maxima) or two successive troughs (minima) Unit 7 (Part I) - Wave Motion 8

9 (2) Amplitude (A) The distance from any point on the wave from its undisturbed position (i.e., the midpoint) is called the displacement (x). The maximum possible displacement is called the amplitude (A) of a wave. Both quantities are usually measured in meters. Unit 7 (Part I) - Wave Motion 9

10 (3) Frequency (f) and period (T) Frequency of a wave is the number of vibrations per second. It is measured in the unit of Hertz (Hz). The time taken for one complete vibration is called the period, measured in the unit of seconds. These two quantities have the inverse relationship: f = 1 T Unit 7 (Part I) - Wave Motion 10

11 Example: A dog s tail wags 50.0 times in 40.0 s. (a) What is the frequency of the tail? (b) What is the period of vibration of the tail? (a) The frequency of the tail is f = = 1.25 Hz (b) The period of vibration of the tail is T = 1 f = = 0.80 s Unit 7 (Part I) - Wave Motion 11

12 Example: A certain tuning fork makes 7680 vibrations in 30 s. (a) What is the frequency of the tuning fork? (b) What is the period of the tuning fork? (a) The frequency of the tuning fork is f = (b) The period of the tuning fork is = 256 Hz T = 1 f = = s Unit 7 (Part I) - Wave Motion 12

13 Another important quantity that is used to describe a wave is velocity. Velocity of a wave refers to the velocity at which the wave crests (or any other part of the wave) move. It is also called the phase velocity. This, however, is not the same as the velocity of the particles in the medium. Unit 7 (Part I) - Wave Motion 13

14 From mechanics, velocity is defined as v = d t In a wave motion, the time taken for a complete wave to go is the period T, while the distance travelled by a complete wave is wavelength λ. Hence, v = λ T Since 1 T = f, we can rewrite the equation above: v = fλ This equation is called the general wave equation. Unit 7 (Part I) - Wave Motion 14

15 Example: What is the speed of a sound wave if its frequency is 256 Hz and its wavelength is 1.29 m? The speed of sound is v = fλ = = 330 m/s What is the frequency of a sound wave if its speed is 340 m/s and its wavelength is 1.70m? The frequency of sound is f = v λ = = 200 Hz Unit 7 (Part I) - Wave Motion 15

16 Example: A hiker shouts toward a vertical cliff 685 m away. The echo is heard 4.00 s later. The wavelength of the sound is m. (a) What is the speed of sound in air? (b) What is the frequency (c) What is the period of the wave? (a) v = d t = = 343 m/s (b) f = v λ = = 457 Hz (c) T = 1 f = = s Unit 7 (Part I) - Wave Motion 16

17 2. Behaviors of waves in one dimension So far we have only looked at the properties that waves possess in general, and we have implicitly assumed that the propagation of waves is continuous throughout the medium and there exists no boundary. However, if a wave travels through a medium and hits a boundary (e.g. an end of a cord), then several things will happen. First of all, the speed and wavelength of a wave change when it moves from one medium to another. However, frequency remains the same. Unit 7 (Part I) - Wave Motion 17

(1) Transmission and reflection At the boundary between two media, a partial reflection of wave occurs in which some energy is transmitted into the new medium while some is reflected back into the

18 (1) Transmission and reflection At the boundary between two media, a partial reflection of wave occurs in which some energy is transmitted into the new medium while some is reflected back into the original medium. The transmitted wave retains its phase in the partial reflection, but the phase of the wave may change depending on the relative change of speed between two media. The speed of wave travelling in a medium depends on its density. The lower the density, the faster the wave moves. Speed of wave on a string (in m/s) v = T ρ Linear density of the string (in kg/m) Unit 7 (Part I) - Wave Motion 18

19 If a wave is traveling from a light string to a heavy string, speed decreases and the reflected wave will be inverted. Both the transmitted and reflected waves are shorter than the original wave. If a wave is traveling from a heavy string to a light string, speed increases and the reflected wave will be upright (or erected). Both the transmitted and reflected waves are shorter than the original wave. Unit 7 (Part I) - Wave Motion 19

20 Same idea can be used to understand the phase changes observed in the following situations. String is connected to a wall (more dense) Speed decreases when passing the boundary The reflected wave is therefore inverted. String is connected to a free end (less dense) Speed increases when passing the boundary The reflected wave hence remains upright Unit 7 (Part I) - Wave Motion 20

individually. The additive property of waves is called the principle of superposition.

21 (2) Interference of waves When two waves meet, they interact with one another, resulting in an interference. The resultant displacement of a given particle is equal to the sum of the displacements produced by each wave individually. The additive property of waves is called the principle of superposition. Once the interference is over, the waves pass through each other unaffected and are not reflected. Unit 7 (Part I) - Wave Motion 21

If two or more waves of different wavelengths act simultaneously, the resultant waveform

22 When two waves interfere to produce a resultant wave which has a greater displacement than those of the original waves, it is called a constructive interference. If two or more waves of different wavelengths act simultaneously, the resultant waveform will be determined by the algebraic sum of all the individual wave displacements. Unit 7 (Part I) - Wave Motion 22

There is an interesting feature when two waves of same amplitude interact destructively.

23 Similarly, when the displacement of the resultant wave in an interference is smaller than those of the individual waves, it is called a destructive interference. There is an interesting feature when two waves of same amplitude interact destructively. During the interference, the two waves cancel out each other, giving rise to a zero resultant displacement. Apparently, the waves disappear! Unit 7 (Part I) - Wave Motion 23

24 In general, for two waves of the same wavelength but different amplitudes, the constructive and destructive interferences result in the following patterns respectively. Constructive interference Destructive interference Unit 7 (Part I) - Wave Motion 24

25 For the case where waves of different wavelengths and amplitudes are interfering one another, the resultant waveform may be very complex. For examples, mixing sound waves of different frequencies produces beat. Unit 7 (Part I) - Wave Motion 25

be able to resolve the waveform and determine what

26 Fortunately, by the aid of a mathematical trick (called Fourier transform) and computers, we would be able to resolve the waveform and determine what are the waves involved. Unit 7 (Part I) - Wave Motion 26

27 The following is a snapshot from a computer program that allows for the calculations using Fourier transform. (The program Fourier Series 3D by Tomáš Bořil.) Unit 7 (Part I) - Wave Motion 27

(3) Standing waves When two waves of same wavelength and amplitude traveling in opposite directions interfere, a standing wave interference pattern, or simply standing wave, is formed.

28 (3) Standing waves When two waves of same wavelength and amplitude traveling in opposite directions interfere, a standing wave interference pattern, or simply standing wave, is formed. Standing wave is peculiar in a way that there are points, called nodal point N, that remain at rest throughout the interference. The midpoint between two consecutive nodal points is an antinode at which double crests and double trough occur. Unit 7 (Part I) - Wave Motion 28

29 In standing waves, the distance between two successive nodes is half of the wavelength. Example: The distance between two successive nodes in a vibrating string is 10 cm. The frequency of the source is 30 Hz. What is the wavelength of the waves? What is their velocity? The wavelength is: The velocity is: λ = = 0.20 m v = fλ = = 6 m/s Unit 7 (Part I) - Wave Motion 29

30 The frequencies at which standing waves can exist in a rope of a given length whose ends are fixed are the natural frequencies, or resonant frequencies, of the rope. The lowest resonant frequency possible is called the fundamental. The other higher resonant frequencies are called overtones. Unit 7 (Part I) - Wave Motion 30

31 For a string which is fixed at two ends, it can only vibrate in certain patterns, always with nodes at each end. The resonant frequencies are therefore whole-numbered multiples of the fundamental frequency. The frequencies of the harmonics (or overtones) are given by f n = nf 0 where n is the order of the harmonic and f 0 is the fundamental frequency. Unit 7 (Part I) - Wave Motion 31

32 3. Behaviors of waves in two dimensions Depending on the nature of the source, the wave generated may be of various shapes. For example, while a point source creates circular waves, a linear source produces plane waves. A continuous crest or trough is called a wavefront. It also represents the equal number of wavelengths from the source of the waves. Unit 7 (Part I) - Wave Motion 32

33 For a constant frequency source, if the speed does not change, the wavelength of the wave remains the same. It implies that the spacing between successive wavefronts also remains the same. This is dictated by the wave equation. To show the direction of motion of the wavefronts, arrows are drawn at right angle to the wavefronts from the source of wave. These arrows are called wave rays. Unit 7 (Part I) - Wave Motion 33

34 To study the behaviors of waves in two dimensions, a specially designed device called ripple tank is used. Unit 7 (Part I) - Wave Motion 34

$In general, waves in 2-D show the following behaviors as the 1-D counterparts: (1) reflection (2) interference plus something new: (3) refraction (4) diffraction A 3D plot of a 2D wave interference.$

35 Similar to waves propagating in one-dimension along a string or a rope, waves transmitting in two-dimension demonstrate some unique features. In general, waves in 2-D show the following behaviors as the 1-D counterparts: (1) reflection (2) interference plus something new: (3) refraction (4) diffraction A 3D plot of a 2D wave interference. From: Unit 7 (Part I) - Wave Motion 35

(1) Reflection When a wave runs into an obstacle, it gets reflected. For instance, ocean waves crashing onto a rocky cliff are reflected away from the cliff.

36 (1) Reflection When a wave runs into an obstacle, it gets reflected. For instance, ocean waves crashing onto a rocky cliff are reflected away from the cliff. Sound waves reflected from a concrete wall are heard as echoes. While the reflection of wave in 1D is straightforward, that in 2D is more complex. The way that a wave is reflected depends on the angle of incidence and the shape of the obstacle. Unit 7 (Part I) - Wave Motion 36

When plane waves run into a straight barrier, they are

reflection, which is equal to the angle of incidence

37 When plane waves run into a straight barrier, they are reflected along their original path. When a plane wave hits a straight barrier obliquely, the wavefront is reflected at an angle, called angle of reflection, which is equal to the angle of incidence relative to the normal of the barrier. θ i = θ r Unit 7 (Part I) - Wave Motion 37

If the reflector is curved, the plane waves will be reflected to one point called the focal point. This can be understood by considering the laws of reflection and wave rays.

38 If the reflector is curved, the plane waves will be reflected to one point called the focal point. This can be understood by considering the laws of reflection and wave rays. A concave surface converges the reflected wavefronts to the focal point in front of the surface. A convex surface, on the other hand, diverges the wavefronts from an imaginary focal point behind the barrier. Unit 7 (Part I) - Wave Motion 38

39 The patterns of reflection of a circular wave are quite similar to those of plane waves. The reflection of circular waves on a concave surface: Source beyond the focal point Reflected waves converge Source at the focal point Reflected waves are planar Source within the focal length Reflected waves diverge Unit 7 (Part I) - Wave Motion 39

40 The reflection of circular waves on other types of surface: The surface is straight The reflected waves are circular, diverging from an imaginary image behind the surface The surface is convex The reflected waves are always diverging from an imaginary image behind the surface Unit 7 (Part I) - Wave Motion 40

41 (2) Refraction It has been noticed that when a wave of a constant frequency is traveling from one medium to another, its speed and wavelength change. The relationship between a wave traveling in different media can be deduced from the following: v 1 v 2 = f 1λ 1 f 2 λ 2 = λ 1 λ 2 because f 1 = f 2 Unit 7 (Part I) - Wave Motion 41

According to fluid mechanics, it is found that the speed of surface water wave depends on both

42 When a wave travels from deep water to shallow water, in such a way that it meets the boundary between the two depths straight on, no change in direction occurs. However, its wavelength decreases. According to fluid mechanics, it is found that the speed of surface water wave depends on both wavelength, λ, and the depth of water, h. If h > λ: v λ If h λ: v h Side view Top view Unit 7 (Part I) - Wave Motion 42

43 On the other hand, if a wave meets the boundary at an angle, the direction of travel does change in addition to its wavelength. When a wave travels at an angle into a medium in which its speed decreases, the refracted wave ray is bent towards the normal of the boundary. When the wave travels at an angle into a medium in which its speed increases, the refracted wave ray is bent away from the normal. θ i > θ R θ i < θ R Unit 7 (Part I) - Wave Motion 43

$To understand the reason of the refraction of$ analogies.

44 To understand the reason of the refraction of waves, we can make use of the following analogies. Case 1: Marching troops Case 2: Toy car rolling onto carpets Unit 7 (Part I) - Wave Motion 44

$A profound effect of refraction of waves is manifested by coastal waves.$ It causes the concentration of energy at the headlands leading to erosion of the cliffs.

It causes the concentration of energy at the headlands leading to erosion of the cliffs.

45 A profound effect of refraction of waves is manifested by coastal waves. Water waves bend as they reach nearshore zone due to different water depth. It causes the concentration of energy at the headlands leading to erosion of the cliffs. On the other hand, it spreads energy in bay areas, resulting in deposition of sediment and the formation of quiet beaches. Unit 7 (Part I) - Wave Motion 45

46 (3) Interference Similar to the 1-D situation, waves moving in two dimensions can interfere either constructively or destructively. When two sources are vibrating in phase, the circular wavefronts move toward one another and interfere, sometimes crest to crest, trough to trough, or crest to trough. Crest/crest or trough/trough interference leads to an antinode, while crest/trough interference leads to a node. Unit 7 (Part I) - Wave Motion 46

47 These give rise to areas of constructive interference and destructive interference alternating in symmetrical manner. Unit 7 (Part I) - Wave Motion 47

48 Some snapshots of the interference patterns. Unit 7 (Part I) - Wave Motion 48

49 (4) Diffraction This is a very interesting phenomenon when a wave encounters an obstacle or a slit whose size is comparable to its wavelength. The wave bends around the corners of the obstacle and changes its wave form. Diffraction of waves can be explained by the Huygens-Fresnel principle and the principle of superposition. (We won t go through the details in Physics 11!) It is observed that the patterns of diffraction depend on: (i) the width of the slit (ii) the shape of the obstacle. Unit 7 (Part I) - Wave Motion 49

$Common diffraction patterns observed for waves passing through a slit: When the slit is widened, the$

50 Common diffraction patterns observed for waves passing through a slit: When the slit is widened, the diffraction pattern changes from circular (c), semi-circular (b), to planar (a). Unit 7 (Part I) - Wave Motion 50

$The amount of diffraction when a wave is passing an obstacle depends on the wavelength of the wave and the$ If the wavelength is much large than the object, the wave will bend around it as if it were not there.

If the wavelength is much large than the object, the wave will bend around it as if it were not there.

51 The amount of diffraction when a wave is passing an obstacle depends on the wavelength of the wave and the size of the obstacle. If the wavelength is much large than the object, the wave will bend around it as if it were not there. However, when the wavelength decreases, a shadow region starts to appear behind the object, in which no diffraction enters into. Shorter the wavelength, larger the shadow region. Unit 7 (Part I) - Wave Motion 51

$The effect of diffraction of waves is manifested in the transmission of microwave and radio-wave TV signals in a hilly region.$

52 The effect of diffraction of waves is manifested in the transmission of microwave and radio-wave TV signals in a hilly region. Microwave has a higher frequency than radio wave; therefore its wavelength is smaller. When passing over a mountain, radio waves are diffracted more, and the antenna on the roof of a house behind the hill can receive them. On the other hand, microwaves are not much diffracted, and thus no signal can be received by the antenna. (In other words, the house is in the shadow region!) Unit 7 (Part I) - Wave Motion 52

Mechanical waves Electromagnetic waves

Mechanical waves Electromagnetic waves Waves Energy can be transported by transfer of matter. For example by a thrown object. Energy can also be transported by wave motion without the transfer of matter. For example by sound waves and electromagnetic